There are mainly two settings in practice where we use PH to obtain a topological fingerprint of a dataset. The first one is the point cloud setting, where the dataset comes as a point cloud in an ambient space \(\mathbb {R}^n\) . Then, we define PH by constructing a sequence of simplicial complexes induced by the pair-wise distances of data points (Vietoris-Rips filtration) and keeping track of the topological changes in this sequence [1]}, [2]}. The second one is the network setting where the typical PH construction uses a filtering function on the network. By construction, while the principal identifier to define PH in the point cloud setting is the pair-wise distances of points, the principal identifier in the network setting is the filtering function. Because of this, PH machinery works differently in a network setting, as explained in Section .
